Merentes Uniformly continuous composition operators in the space of functions of two variables of bounded ϕ-variation in the sense of Wiener Abstract

J. A. Guerrero, J. Matkowski, N. Merentes

Uniformly continuous composition operators in the space of functions of two variables of bounded

ϕ-variation in the sense of Wiener

Abstract. Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of boun- ded Wiener ϕ-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable.

2000 Mathematics Subject Classification: 47B33, 26B30, 26B40.

Key words and phrases: ϕ-variation in the Wiener sense, composition operator, uni- formly continuous operator, left-left regularization, Jensen equation.

1. Introduction. Let I_a^bdenote the rectangle [a₁, b1]×[a2, b2] with the vertices a = (a1, a2), b = (b₁, b2) ∈ ℝ². Let (X, | · |), (Y, | · |) be real normed spaces and C be a set in X. For a function h : I_a^b× C −→ Y define the Nemytskii composition operator H : C^Iab−→ Y^Iab by

H(f )(t, s) = h(t, s, f (t, s)), f ∈ C^Iab, (t, s)∈ Ia^b,

where C^Iab stands for the family of all functions f : I_a^b −→ C. The function h is called the generator of the composition operator H.

Let (BVϕ(I_a^b, X),k · k^ϕ) be the Banach space of functions f ∈ X^Iab which are of bounded ϕ-variation in the sense of Wiener [9], where the norm k · k^ϕ is defined with the aid of Luxemburg-Nakano-Orlicz seminorm [4, 7, 8].

Assume that H maps the set of functions f ∈ BV^ϕ(I_a^b, X) such that f (I_a^b) ⊂ C into BVψ(I_a^b, Y ). In the present paper we prove that, under some conditions, if H is uniformly continuous, then the left-left, right-right, left-right and right-left

regularizations of its generator h with respect to the two first variables are affine functions in the third variable.

This generalizes the result of Chistyakov [2] where it is assumed that H is Lipschitzian.

2. Preliminaries. We begin this section with some notations.

Let a = (a1, a2), b = (b1, b2) ∈ ℝ² be such that ai< bi, i = 1, 2. In the sequel I_a^b := [a1, b1] × [a², b2] denotes the basic rectangle. Let ξ = (ti)^m_i=0 and η = (sj)ⁿ_j=0 be partitions of [a1, b1] and [a2, b2], respectively (i.e., m, n ∈ ℕ, a¹ = t0 < t1 <

· · · < t^m= b₁ and a₂ = s₀< s1 <· · · < sⁿ = b₂). For each function f ∈ X^Iab, we define

∆₁₀f (ti, sj) = f(t_i, sj) − f(ti−1, sj)

∆₀₁f (ti, sj) = f(ti, sj) − f(tⁱ, sj−1)

∆₁₁f (ti, sj) = f(ti−1, sj−1) − f(tⁱ−1, sj) − f(tⁱ, sj−1) + f(ti, sj).

We denote by F the set of all non-decreasing continuous functions ϕ : [0, +∞) −→

[0, +∞) such that ϕ(t) = 0 if and only if t = 0, and limt−→∞ϕ(t) =∞.

Definition 2.1 (Chistyakov [2])

Let ϕ ∈ F, X be a real normed space and f ∈ X^Iab be a function.

(i) For x₂ ∈ [a2, b2], the Jordan ϕ-variation of the function f(·, x2) in [x₁, y1] ⊂ [a₁, b1], denoted v_ϕ,[x1,y1](f) = v_ϕ,[x1,y1](f(·, x2)), is defined by

(1) v_ϕ,[x1,y1](f) := sup

ξ

Xm i=1

ϕ (|∆¹⁰f (ti, x2)|) ,

where the supremum is taken over all the partitions ξ = (ti)^m_i=0 of [x1, y1].

(ii) For x1 ∈ [a¹, b1], the Jordan ϕ-variation of the function f(x1,·) in [x², y2] ⊂ [a2, b2] is defined by

(2) vϕ,[x₂,y₂](f) := sup

η

Xn j=1

ϕ (|∆⁰¹f (x1, sj)|) ,

where the supremum is taken over all the partitions η = (sj)ⁿ_j=0 of [x2, y2].

(iii) The (two-dimensional) Hardy-Vitali-Wiener ϕ-variation of a function f ∈ X^Iab, denoted v_ϕ,Ia^b(f), is defined by

(3) vϕ,I_a^b(f) := sup

(ξ,η)

Xm i=1

Xn j=1

ϕ (|∆¹¹f (ti, sj)|) ,

where the supremum is taken over all pairs m, n ∈ ℕ and (ξ, η) with ξ a partition of [a1, b1] and η a partition of [a2, b2] of the above form.

(iv) The Wiener total ϕ-variation of f ∈ X^Iab is defined by

(4) T Vϕ(f) = T Vϕ(f, I_a^b) := v_ϕ,[a1,b₁](f(·, a²)) + v_ϕ,[a2,b₂](f(a1,·)) + vϕ,I_a^b(f).

(v) A function f ∈ X^Iab is of bounded Wiener total ϕ-variation in I_a^b, if T Vϕ(f) <

∞.

(vi) The class of all functions f ∈ X^Iba with finite Wiener total ϕ-variation is denoted by Vϕ(I_a^b, X); that is

Vϕ(I_a^b, X) :=n

f ∈ X^Iab : T Vϕ(f) < ∞o .

We denote by BVϕ(I_a^b, X) the vector space generated by Vϕ(I_a^b, X), i.e.

BVϕ(I_a^b, X) =n

f ∈ X^Iab : ∃ λ > 0 such that λf ∈ V^ϕ(I_a^b, X)o . In the space BVϕ(I_a^b, X) we define the norm

kfk^ϕ:= |f(a)| + p^ϕ(f), where

pϕ(f) = inf { > 0 : T V^ϕ(f/) ¬ 1} . Some properties of pϕare in the following lemma.

Lemma 2.2 (Chistyakov [2]) For f ∈ BV^ϕ(I_a^b, X), we have (a) if (t, s), (t⁰, s⁰) ∈ Ia^b, then |f(t, s) − f(t⁰, s⁰)| ¬ 4ϕ⁻¹(1)pϕ(f);

(b) if pϕ(f) > 0 then T Vϕ

 _f

pϕ(f)

¬ 1;

(c) if λ > 0 then

(c1) p_ϕ(f) ¬ λ if and only if T Vϕ

f λ

¬ 1;

(c2) if T Vϕ(f/λ) = 1 then pϕ(f) = λ.

Theorem 2.3 (Chistyakov [2]) If ϕ ∈ F is convex and X is a Banach space, then BVϕ(I_a^b), k · k^ϕ
is a Banach space.

In the sequel we will use the notions of left-left regularization and the left- left continuity of a function of two variables. Let f : I_a^b −→ X. If the function f⁻ : I_a^b−→ X given by

f⁻(x1, x2) :=



























lim_y1→x− 1

y2→x⁻₂

f (y1, y2) for (x1, x2) ∈ (a¹, b1] × (a², b2], lim_y1→x−

1

y₂→a⁺₂

f (y1, y2) for x1∈ (a¹, b1] and x2= a2, lim_y1→a+

1

y2→x⁻₂

f (y1, y2) for x1= a1 and x2∈ (a², b2], lim_y1→a⁺

1

y2→a⁺₂

f (y1, y2) for x1= a1 and x2= a2.

is well defined, that is, if all the above limits exist, then f⁻ is called the left-left regularization of f.

Remark 2.4

It is to be noted that (y1, y2) → (x⁻1, x⁻₂) means that (y1, y2) ∈ Ia^b, yi< xi, i = 1, 2, and (y1, y2) → (x¹, x2) inℝ², and similarly for the other limits.

Remark 2.5

In a similar way we can define the right-right, left-right, right-left regularizations of a function f ∈ BV^ϕ(I_a^b, X).

Definition 2.6 A function f : I_a^b−→ X is said to be left-left continuous if

(x,y)→(tlim⁻,s⁻)f (x, y) = f (t, s) for all (t, s) ∈ (a¹, b1] × (a², b2].

Lemma 2.7 (Chistyakov [2]) If f ∈ BV^ϕ(I_a^b, X) then f⁻exists, f⁻ ∈ BVϕ⁻(I_a^b, X) and f⁻ is left-left continuous.

Remark 2.8

The respective counterparts of Lemma 2.7 for the right-right, left-right and right-left regularizations hold also true.

We denote by BV_ϕ⁻(I_a^b, X) the subspace of BVϕ(I_a^b, X) of those functions which are left-left continuous on (a₁, b1] × (a2, b2].

3. Main result. Denote by A(X, Y ) the space of all additive mappings A : X −→ Y and by L(X, Y ) the space of all continuous linear mappings A : X −→ Y .

The main result of this section reads as follows:

Theorem 3.1 Let I_a^b ⊂ ℝ² be rectangle, (X, | · |) be a real normed space, (Y, | · |) be a real Banach space, C be a closed convex subset in X and assume that ϕ, ψ ∈ F. If the composition operator H generated by h : Ia^b× C −→ Y maps BVϕ(I_a^b, C) into BVψ(I_a^b, Y ), and is uniformly continuous, then there exist functions A : I_a^b−→ A(X, Y ) and B : Ia^b−→ Y such that

h⁻(t, s, x) = A(t, s)x + B(t, s), (t, s) ∈ Ia^b, x∈ C,

where h⁻ is the left-left regularization of the function (t, s) → h(t, s, x). Moreover, if 0 ∈ C and intC 6= ∅, then A : Ia^b−→ L(X, Y ) and B ∈ BVψ⁻(I_a^b, Y ).

Proof For every x ∈ C, the constant function Ia^b 3 (t, s) −→ x belongs to BVϕ(I_a^b, C). Since H maps BVϕ(I_a^b, C) into BVψ(I_a^b, Y ), the function I_a^b 3 (t, s) 7→

h(t, s, x) belongs to BVψ(I_a^b, Y ). Now Lemma 2.7 implies the existence of the left- left regularization h⁻ of function (t, s) → h(t, s, x). By assumption, H is uniformly continuous on BVϕ(I_a^b, C). Let ω : ℝ⁺ −→ ℝ⁺ be the modulus continuity of H, that is

ω(ρ) := sup

kH(f1) − H(f2)k^ψ: kf1− f2k^ϕ¬ ρ; f1, f₂∈ BV^ϕ(Ia^b, C)

, for ρ > 0.

Hence we get

(5) kH(f¹) − H(f²)k^ψ ¬ ω (kf¹− f²k^ϕ) , for f1, f2∈ BV^ϕ(I_a^b, C).

From the definition of the norm k · k^ψ we obtain

(6) pψ(H(f1) − H(f²)) ¬ kH(f¹) − H(f²)k^ψ, for f1, f2∈ BV^ϕ(I_a^b, C).

From Lemma 2.2(c1) and (6), if ω (kf¹− f²k^ϕ) > 0, then (7) vψ,[a₁,b₁]

(H(f1) − H(f²)) (·, a²) ω (kf¹− f²k^ϕ)



¬ T V^ψ

H(f1) − H(f²) ω (kf¹− f²k^ϕ)



¬ 1.

Therefore, for any a1 = α1 < β1 < α2 < β2 <· · · < α^m< βm= b1; a2 = α1 <

β₁< α2 < β₂<· · · < α^m< β_m = b2, m∈ ℕ, the definitions of the operator H and the functional vψ,[a1,b1](·, a²), imply that

(8)

Pm

i=1ψ_|h(α

i,αi,f1(αi,αi))−h(αⁱ,αi,f2(αi,αi))−h(αⁱ,βi,f1(αi,βi))+h(αi,βi,f2(αi,βi)) ω(kf1−f2kϕ)

−h(βi,αi,f1(βi,αi))+h(βi,αi,f2(βi,αi))+h(βi,βi,f1(βi,βi))−h(βi,βi,f2(βi,βi))|

ω(kf1−f2kϕ)



¬ 1.

For α, β ∈ ℝ, α < β, we define functions η^α,β :ℝ −→ [0, 1] by

(9) ηα,β(t) :=







0 if t ¬ α

t−α

β−α if α ¬ t ¬ β 1 if β ¬ t .

We first fix t ∈ (a¹, b1], s ∈ (a², b2], m ∈ ℕ. For arbitrary finite sequence a¹< α1<

β1< α2< β2<· · · < α^m< βm< t, a2< α1< β₁< α2< β₂<· · · < α^m< β_m<

s and x1, x2∈ C, x¹6= x², the functions f1, f2: I_a^b−→ X defined by (10)

f`(τ, γ) := 1 2

hηαi,βi(τ) + η_α

i,β_i(γ) − 1

(x1− x²) + x`+ x2

i, (τ, γ)∈ Ia^b, ` = 1, 2,

belong to BVϕ(I_a^b, C). From (10) we have

f1− f²=x1− x²

2 ,

therefore

kf¹− f²k^ϕ=  x1− x²

2   ; and, moreover,

f1(α_i, ¯αi) = x₂; f₂(α_i, ¯αi) = −x1+ 3x₂

2 ; f₁(β_i, ¯αi) = x₂; f₂(β_i, ¯αi) = −x1+ 3x₂

2 ;

f1(βi, ¯βi) = x1; f2(βi, ¯βi) = x1+ x2

2 ; f1(αi, ¯βi) =x1+ x2

2 ; f2(αi, ¯βi) = x2. Using (8), we hence get

(11)

Xm i=1

ψ

h(αi, αi, x₂) − h(αⁱ, αi,^−x1+3x₂ ²) − h(αⁱ, βi,^x1+x₂ ²) + h(αi, βi, x₂) ω(kf1− f2k^ϕ)

−h(βi, αi, x2) + h(βi, αi,^−x1+3x₂ ²) + h(βi, βi, x1) − h(βi, βi,^x1+x₂ ²) ω(kf¹− f2kϕ)

!

¬ 1.

Since, for any x ∈ C, the constant function Ia^b3 (t, s) −→ x belongs to BV^ϕ(I_a^b, C) and H maps BVϕ(I_a^b, C) into BVψ(I_a^b, Y ), the function I_a^b 3 (t, s) 7→ h(t, s, x) is in BVψ(I_a^b, Y ) for any fixed x∈ C. From continuity of ψ and the left-left continuity of h⁻ (Lemma 2.7), letting (α1, α1) tend to (t, s) from the left in (11), we obtain

Xm i=1

ψ





h⁻(t, s, x1) − 2h⁻ t, s,^x1+x₂ ²
+ h⁻(t, s, x2) ω_|x

1−x2| 2





 ¬ 1,

that is ψ





h⁻(t, s, x₁) − 2h⁻ t, s,^x1+x₂ ²
+ h⁻(t, s, x₂) ω_|x

1−x2| 2





 ¬ 1 m. Hence, since m ∈ ℕ is arbitrary,

ψ





h⁻(t, s, x1) − 2h⁻ t, s,^x1+x₂ ²
+ h⁻(t, s, x2) ω

|x1−x2| 2





 = 0.

As ψ ∈ F, we obtain

h⁻(t, s, x₁) − 2h⁻

t, s,x1+ x₂ 2



+ h⁻(t, s, x₂) = 0.

Therefore

(12) h⁻



t, s,x1+ x2

2



= h⁻(t, s, x1) + h⁻(t, s, x2) 2

for all (t, s) ∈ (a¹, b1] × (a², b2], and for all x1, x2∈ C.

For t ∈ (a¹, b1] and s = a2, let us fix a1 < α1 < β1 < α2 < β2 <· · · < α^m <

βm< t and a2< α1< β₁ < α2< β₂<· · · < α^m< β_m< b2. Proceeding as above we get (11). Taking the limit as (α1, β_m) → (t⁻, a⁺₂) and by (11), we get, again (12). The cases when t = a1 and s ∈ (a², b2] or t = a1 and s = a2 can be treated similarly. Consequently,

h⁻



t, s,x1+ x2

2



= h⁻(t, s, x1) + h⁻(t, s, x2) 2

is valid for all (t, s) ∈ Ia^b and all x1, x2∈ C.

Therefore, the function h⁻(t, s, ·) satisfies the Jensen functional equation in C, for each (t, s) ∈ Ia^b. Adapting the standard argument (cf. Kuczma [3]), we conclude that, for each (t, s) ∈ Ia^b there exist an additive function A(t, s) and B(t, s) ∈ Y such that

(13) h⁻(t, s, x) = A(t, s)x + B(t, s), x∈ C, (t, s) ∈ Ia^b.

The “moreover part”, follow from the uniform continuity of the operator H : BVϕ(I_a^b, C) −→ BV^ψ(I_a^b, Y ) and intC 6= ∅ imply the continuity of the func- tion A(t, s), consequently A(t, s) ∈ L(X, Y ).

Since 0 ∈ C, putting x = 0 in (13), we get

h⁻(t, s, 0) = B(t, s), (t, s) ∈ Ia^b,

which shows that B ∈ BV^ψ(I_a^b, Y ). _■

Remark 3.2 The counterparts of Theorem 3.1 for the right-right, right-left and left-right regularizations are also valid.

Remark 3.3 The uniformly continuous composition operators for functions of bounded variation in a single variable were considered in [5].

References

[1] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operator, Cambridge University Press, New York, 1990.

[2] V. V. Chistyakov, Superposition Operators in the Algebra of Functions of two Variables with Finite Total Variation, Monatshefte f¨ur Mathematik 137 (2002), 99-114.

[3] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Editors and Silesian University, Warszawa -Kraków- Katowice, 1985.

[4] W. A. Luxemburg, Banach Function Spaces, Ph.D. thesis, Technische Hogeschool te Delft, Netherlands, 1955.

[5] J. Matkowski, Uniformly continuous superposition operators in the space of bounded variation functions, Math. Nachr.282 (2010).

[6] J. Matkowski and J. Miś, On a Characterization of Lipschitzian Operators of Substitution in the Space BV ([a, b]), Math. Nachr.117 (1984), 155-159.

[7] H. Nakano, Modulared Semi-Ordered Spaces, Tokyo, 1950.

[8] W. Orlicz, A note on modular spaces. I, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom.

Phys.9 (1961), 157-162.

[9] N. Wiener, The quadratic variation of function and its Fourier coefficients, Massachusetts J.

Math.3 (1924), 72-94.

J. A. Guerrero

Universidad Nacional Experimental del T´achira, Dpto. de Matem´aticas y F´ısica San Cristóbal-Venezuela

E-mail: jaguerrero4@gmail.com, jguerre@unet.edu.ve J. Matkowski

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra Zielona Góra, Poland

Institute of Mathematics, Silesian University Katowice, Poland

E-mail: J.Matkowski@wmie.uz.zgora.pl N. Merentes

Universidad Central de Venezuela, Escuela de Matem´aticas Caracas-Venezuela

E-mail: nmer@ciens.ucv.ve

(Received: 28.09.2009)